# Isolation of cuspidal spectrum, with application to the Gan–Gross–Prasad conjecture

@article{BeuzartPlessis2020IsolationOC,
title={Isolation of cuspidal spectrum, with application to the
author={Raphael Beuzart-Plessis and Yifeng Liu and Wei Zhang and Xinwen Zhu},
journal={arXiv: Number Theory},
year={2020}
}
• Published 16 December 2019
• Mathematics
• arXiv: Number Theory
We introduce a new technique for isolating components on the spectral side of the trace formula. By applying it to the Jacquet--Rallis relative trace formula, we complete the proof of the global Gan--Gross--Prasad conjecture and its refinement Ichino--Ikeda conjecture for $\mathrm{U}(n)\times\mathrm{U}(n+1)$ in the stable case.
Regularized Periods and the global Gan-Gross-Prasad conjecture : The case of $U(n+2r) \times U(n)$
In this paper, we introduce regularized trilinear periods on certain non-reductive groups. It has two direct applications. Firstly, it enables us to define the regularized Bessel periods and the
The global Gan-Gross-Prasad conjecture for unitary groups: the endoscopic case
• Mathematics
Publications mathématiques de l'IHÉS
• 2022
In this paper, we prove the Gan-Gross-Prasad conjecture and the Ichino-Ikeda conjecture for unitary groups U n × U n + 1 $U_{n}\times U_{n+1}$ in all the endoscopic cases. Our main technical
The Fourier--Jacobi periods : The case of $U(n+2r) \times U(n)$
The global Gan-Gross-Prasad conjecture predicts that the non-vanishing of a certain period is equivalent to the non-vanishing of the central value of a certain Rankin-Selberg $L$-function. There are
On the Beilinson–Bloch–Kato conjecture for Rankin–Selberg motives
• Mathematics
Inventiones mathematicae
• 2022
In this article, we study the Beilinson-Bloch-Kato conjecture for motives corresponding to the Rankin-Selberg product of conjugate self-dual automorphic representations, within the framework of the
Holomorphy of adjoint L-functions for GL(n): $$n\le 4$$
We show entireness of complete adjoint L-functions associated to \textbf{any} cuspidal representations of $\GL(3)$ or $\GL(4)$ over an arbitrary global field. Twisted cases are also investigated.
Spectral aspect subconvex bounds for ${\rm U}_{n+1} \times {\rm U}_{n}$.
Let $(\pi,\sigma)$ traverse a sequence of pairs of cuspidal automorphic representations of an anistropic unitary Gan--Gross--Prasad pair $({\rm U}_{n+1},{\rm U}_n)$ over a number field. We assume
On the Gan-Gross-Prasad conjecture and its refinement for $\left(\mathrm{U}\left(2n\right),\mathrm{U}\left(1\right)\right)$
• Mathematics
• 2022
. We prove the Gan-Gross-Prasad conjecture for ( U ( 2 𝑛 ) , U ( 1 )) in general and prove its reﬁnement, namely the Ichino–Ikeda type explicit formula for the central 𝐿 -values, under certain
1 2 A pr 2 02 2 A spectral expansion for the symmetric space GL n ( E ) / GL n ( F )
In this article we state and prove the spectral expansion of theta series attached to the symmetric space GLn(E)/GLn(F ) where n > 1 and E/F is a quadratic extension of number fields. This is an
Automorphic Forms on Unitary Groups
This manuscript provides a more detailed treatment of the material from my lecture series at the 2022 Arizona Winter School on Automorphic Forms Beyond GL2. The main focus of this manuscript is
Spherical varieties, functoriality, and quantization
We discuss generalizations of the Langlands program, from reductive groups to the local and automorphic spectra of spherical varieties,and to more general representations arising as “quantizations”

## References

SHOWING 1-10 OF 60 REFERENCES
On the global Gan–Gross–Prasad conjecture for unitary groups: Approximating smooth transfer of Jacquet–Rallis
• Hang Xue
• Mathematics
Journal für die reine und angewandte Mathematik (Crelles Journal)
• 2017
Abstract Zhang proved the global Gan–Gross–Prasad conjecture for U ⁡ ( n + 1 ) × U ⁡ ( n ) \operatorname{U}(n+1)\times\operatorname{U}(n) under some local conditions [19]. One of the conditions is
Automorphic period and the central value of Rankin--Selberg L-function
We prove a refinement of the global Gan-Gross-Prasad conjecture proposed by Ichino-Ikeda and N. Harris for unitary groups under some local conditions. We need to assume some expected properties of
AST418 - A local trace formula for the Gan-Gross-Prasad conjecture for unitary groups: the archimedean case
In this paper, we prove, following earlier work of Waldspurger ([Wa1], [Wa4]), a sort of local relative trace formula which is related to the local Gan-Gross-Prasad conjecture for unitary groups over
Fourier–Jacobi periods and the central value of Rankin–Selberg L-functions
In this paper, we propose a conjectural formula, relating the Fourier–Jacobi periods of automorphic forms on U(n)×U(n) and the central value of some Rankin–Selberg L-function. This can be viewed as a
Fourier transform and the global Gan–Gross–Prasad conjecture for unitary groups
By the relative trace formula approach of Jacquet�Rallis, we prove the global Gan�Gross�Prasad conjecture for unitary groups under some local restrictions for the automorphic representations
On the Periods of Automorphic Forms on Special Orthogonal Groups and the Gross–Prasad Conjecture
• Mathematics
• 2010
In this paper, we would like to formulate a conjecture on a relation between a certain period of automorphic forms on special orthogonal groups and some L-value. Our conjecture can be considered as a
The descent map from automorphic representations of GL(n) to classical groups
• Mathematics
• 2011
Certain Residual Eisenstein Series Fourier Coefficients of Gelfand-Graev Type and Fourier-Jacobi Type Jacquet Modules Corresponding to Gelfand-Graev Models Jacquet Modules Corresponding to
A Refined Gross-Prasad Conjecture for Unitary Groups
Let F be a number field, AF its ring of adeles, and let [pi]n and [pi]n+1 be irreducible, cuspidal, automorphic representations of SOn(AF) and SOn+1AF), respectively. In 1991, Benedict Gross and
COMPARISON OF LOCAL RELATIVE CHARACTERS AND THE ICHINO–IKEDA CONJECTURE FOR UNITARY GROUPS
Abstract In this paper, we prove a conjecture of Wei Zhang on comparison of certain local relative characters from which we draw some consequences for the Ichino–Ikeda conjecture for unitary groups.